Loss ppt QC

7/23/2019 Loss ppt QC

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$$: Swiss NSF, Nano Center Basel, EU (RTN), DARPA & ONR, ICORP-JST

Spin Qubits in Quantum Dots

Daniel Loss

Department of Physics

University of BaselSwitzerland


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G. Burkard

B. CoishH.A. EngelV. GolovachD. Klauser

M. TrifD. DiVincenzoC. EguesO. Gywat

M. LeuenbergerJ. LevyF. MeierP. RecherE. Sukhorukov

S. TaruchaB. AltshulerD. AwschalomL. Glazman

L. Kouwenhoven

G. Abstreiter

L. SamuelsonL. VandersypenC. MarcusB. WesterveltC. Bruder

D. BulaevN. BonesteelM.S. ChoiK. Ensslin

A. ImamogluJ. LehmannA. MacDonaldD. Saraga

J. SchliemannC. SchnenbergerD. StepanenkoM. BorhaniR. Hanson

J. ElzermanF. Koppens

Special thanks to many Collaborators:


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Outline

A. Spin qubits in quantum dots

1. Basics of quantum computing and quantum dots

2. Quantum gates: interaction based and measurement based entanglement

B. Spin decoherence in GaAs quantum dots

Spin orbit interaction and spin decay Alternative spin qubits: holes, graphene, nanotubes,... Nuclear spins and hyperfine induced decoherence

classical spin bath vs quantum and non-Markovian behavior

C. Nuclear spin order in 2DEGs (ultra-low temp.)

Kondo lattice in marginal Fermi liquid


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Quantum Information

Classical digital computernetwork of Boolean logic gates, e.g. XOR

bits:

physical implementation:e.g. 2 voltage levels

gate: electronic circuit

Quantum computer

physical implementation:quantum 2-level-system:

quantum gate: unitary transformation(is reversible!)

1,0, =ba

1,10,22=++= ba

1,0

qubits


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Quantum Computing (basics)

basic unit: = state of a quantum two-level systemqubit

"natural" candidate: electron spin

1) prepare N qubits (input)2) apply unitary transformation in 2N-dim. Hilbert space computation

3) measure result (output)

quantum computation:

"quantum" computation faster than "classicalfactoring algorithm (Shor 1994): exp N N2

database search (Grover 1996): N N1/2


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Electron qubit: spin better than charge

natural choice for qubit: spin of electron

12 TT

echspin arg >>

1 ns

due to longer relaxation/decoherence* times

Awschalom et al., 97

Tarucha et al., 02Kouwenhoven et al., 03-07

Abstreiter et al., 04Zumbuhl & Kastner et al., '08

10ns -1s

mesoscopicsFujisawa et al. 03Marcus et al. 01

*)

theory: for single spin in GaAs dot (in principle)

GaAs


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Spin qubits in quantum dots key elements

Initialization 1 electron, low T, high B0

Read-out convert spin to charge

then measure charge

ESR pulsed microwave magnetic field

SWAP exchange interaction

H0 ~ i zi

HJ ~ Jij (t) i j

HRF ~ Ai(t) cos(i t) xi

Coherence spin-orbit interaction

nuclear spins

EZ=

gBB

EZ=

gBB

J(t) J(t)

SL S

R

DL & DiVincenzo, PRA 1998


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Electrical control and detection

Tunable # of electrons

Tunable tunnel barriers

Electrical contacts

A quantum dot as a tunable artificial atom

Confinement

Discrete # charges

Discrete orbitals

Delft group (L. Kouwenhoven & L. Vandersypen)


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C. Marcus et al., PRL 2004

GaAs/AlGaAs Heterostruktur

2DEG 90 nm depth, ns = 2.9 x 1011 cm-2 Temp.: 100 mK


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Double Dot in Carbon Nanotube (SW)

Schnenberger group (Grber et al., Phys. Rev. B 74, 075427 (2006))


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Quantum Computing with Spin-QubitsDL & DiVincenzo, PRA 57 (1998)

H(t)=J(t) SLS

R

SL SR

1. Quantum gates based on exchange interaction:

Spin 1/2 of electron = qubit


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SL SR H(t)=J(t) SLSR

CNOT (XOR) gate

s

dttHi

s duringJeTU

s

0,)(0

)('

=

=

x0

01

a b


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H(t)=J(t) SLSRSL SR

CNOT (XOR) gate

2/12/122 121

SW

Si

SW

SiSi

XOR UeUeeUz

zz

=

:SWU

+ i

SW eU :2/1

switching time: 180 psPetta et al., Science, 2005


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Quantum XOR gate with 'sqrt of swap'

,2/12/1)2/()2/( 121

SW

Si

SW

SiSi

XOR UeUeeUzzz +=

| | | |

| (|+i|) (i |+ |) |

i| (|-i|) (i |- |) -i|

i| | - | -i|

i| i | i| -i|

2/1

SWU

z

Sie 1

2/1

SWU

zSi

e2

2

zSi

e1

2

2

4/ie

2

4/ie

2

4/iie

2

4/iie


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Quantum XOR gate with 'sqrt of swap'

2/1

SWU

| |+i|Square-root-of-swap:

Entanglement is crucial for quantum computing!

entangled state= entangler: product state

= |1

x |2

,2/12/1)2/()2/( 121

SW

Si

SW

SiSi

XOR UeUeeUzzz +=


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Note: Control of exchange interaction J and switching timeneeds to be very precise (1:104)

experimental challenge

CNOT gate without interaction?

Yes: CNOT gates based on measurement:

Linear optics & single-photon detection conditional sign flip (non-deterministic) [1]Full Bell state analyzer & GHZ state deterministic quantum computing [2]

Partial Bell-state (parity) measurements deterministic quantum computing [3]

[1] E. Knill, R. Laflamme and G. J. Milburn, Nature 409, 46 (2001).

[2] D. Gottesman and I.L. Chuang, Nature 402, 390 (1999).[3] C.W.J. Beenakker et al., Phys. Rev. Lett. 93, 020501 (2004).


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CNOT gate can be implemented with two parity gates

Beenakker, Kindermann & DiVincenzo, PRL 04

=


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Deterministic entangler:

P

a

b

c

d

a,b: input armsc,d: output arms

baba

input statein arm a

input statein arm b (ancilla)

( ) ( ) ( ) ( )bababababbaa +++=++

Thus, we get:

Beenakker et al., 2004

Projective measurement: measurement of parity p projects input state into

either parallel output state (p=1) or antiparallel output state (p=0). If p=0, thenapply x(d) on output state get always same final output state in arms c and d.

1, =+ pifdcdc

dcdcdcdc pif +=+ ,0,

dcdcaa ++


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| + || - | even parity Bell state: parallel spins

| + || - |

odd parity Bell state: antiparallel spins

Q: Does scheme exist for electron spins tomeasure parity of Bell states non-destructively?

Measurement-based quantum computing

with spin qubitsEngel & DL, Science 309, 586 (2005)

Advantage:parity measurement is digital (0 or 1) quantum gate is digital


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Double Quantum Dot and QPC

Current IQPC depends on charge state1

odd parity: tunneling

even parity: no tunneling

IQPC

Quantumpoint contact

[1] J.M. Elzerman et al., Nature 430, 431 (2004)

IQPC

IQPC

I

I

L R


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Tunneling transitions (coherent)

Single level picture

Many processes forsmall level spacing

But for>>Uonly transitions (a)

and (d) are relevant

|SL(a)

|SL(b)

|SR(c)

|T0L(d)

|T0L(e)

|T0L(f)

|T0

L

(g)


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Phases due to different Zeeman interaction

during virtual occupation of state LR

|LR vs|LRcorrectable via one-qubit gatessuppression via large tdand fast read-out

Detuning from resonance

increases measurement time

Finite exchangeJ for LL and RR

additional dynamical phasecorrectable via one-qubit operations

Tunneling tStT and/orJLJR

|S and |T0 are distinguishable decoherence!

Imperfections


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Universal Quantum Computing with Parity Gates

Two qubit dots (1 and L), paritymeasurement using reference dot (R)

(1, 1, 0) (0, 2, 0) (0, 0, 2)Transfer back adiabatically to (1,1,0)

1

L

R

P =

Parity gate for spin :(Engel & Loss, Science 05):

CNOT gate can be implemented with twoparity gates (Beenakker et al., PRL 04):

=


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Perform parity gate (some details)

12

R2

QPC

time

1 2 3

p


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a

t

R2

c

R1

QPC 1

QPC 2

time

1 2 3

p1

4 5 6 7 8 9 10

p2

H

H

H

H

m

11

Protocol for CNOT gate (1)

c

t

2121 tcctcUCNOT =

H

Engel and DL, 05


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Protocol for CNOT gate (3): Conditional operations p1, p2, m represent the outcome of the measurements of steps 3, 7, and 10.

Detection of even parity (superposition of |00 and |11) is labeled as 0; odd parity as 1

In step 11, single-qubit gates listed on the rhs are applied to qubits c and t

I stands for identity (do nothing), X for X, Z for Z

p1 p2 m c t

0 0 0 I I0 0 1 I X

0 1 0 Z I

0 1 1 Z X

1 0 0 I X

1 0 1 I I

1 1 0 Z X

1 1 1 Z I


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CNOT-Gate in Topological Quantum Computing via Parity(braiding of non-abelian anyons, Kitaev 03)

Zilberberg, Braunecker,and Loss, PRA 77, 012327 (2008)

Ising anyons, Bravyi 06


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i

i

iBiji

ij

ij StBgSStJHrrrr

+= ))(()(

2-qubit gate'sqrt of swap' (s 100 ps)Petta et al., Science '05

1-qubit gatevia ESR for single spin (s 10 ns)Koppens et al., Nature '06, PRL '07;

via EDSR (T2 10 s), Science '07

Spin-Qubits in Dots are Scalable (?)


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i

i

iBiji

ij

ij StBgSStJHrrrr

+= ))(()(

2-qubit gate'sqrt of swap' (s 100 ps)Petta et al., Science '05

1-qubit gatevia ESR for single spin (s 10 ns)Koppens et al., Nature '06, PRL '07;

via EDSR (T2 10 s), Science '07

Spin-Qubits in Dots are Scalable (?)

T2/

s~ 103 -105 *)

*) Coish & DL, PRB 75 (2007): s for single spin can be < 1ns (via exchange)

i.e. system is scalable


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Spin-Qubits from Electrons

Many more choices for spin qubits:

SL S

Rsimplest spin-qubit:

spin-1/2 of 1 electron == 1,0

'exchange only qubits' DiVincenzo et al., 20003 electrons:

'singlet-triplet' qubits Levy 2002, Taylor et al., 2005

2 electrons: 'spin-cluster qubits' Meier, Levy & DL, `03

N coupled electrons: AF spin chains, ladders, clusters,...

molecular magnets Leuenberger & DL, 01; Affronte et al., 06Lehmann et al., 07; Trif et al., 08

== + 01,0 TTS

01,0 TS ==


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Quantum Computing with Spin-Qubits

SL SR

Need to understand the dynamics and decoherencemechanisms for electron spins in quantum dots

CNOT (XOR) gate based on

entanglement such as

++ ii eore

i.e. phase coherence is crucial


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Spin decoherence in GaAs quantum dots

1) Spin-orbit coupling (Dresselhaus & Rashba)

interaction between spin and charge fluctuations

Relaxation with rate 1/ T1

+ Decoherence with rate 1/ T2= decay of coherent superposition

Two important sources of spin decay in GaAs:

2) Hyperfine interaction between electron spin and nuclear spinsleads to non-exponential decay

G l i H il i


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( )tgH B hSBS +=where h(t) is a fluctuating (internal) field with < h(t) >=0

( ) ( ) ( ) ( )[ ]

+= htEi

YYXX ZethhthhdtT 00Re1

1

( ) ( )

+= thhdtTT ZZ 0Re2

11

12

Relaxation (T1) and decoherence (T2) times in weak coupling:

General spin Hamiltonian:

[if < hi(t) hj(t) >~ij,i,j=(X,Y,Z)]

relaxation

contribution

dephasing

contribution

See e.g. Abragam


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( )tgH B hSBS +=where h(t) is a fluctuating (internal) field with < h(t) >=0

( ) ( )

+= thhdtTT

ZZ 0Re2

11

12

For SOI linear in momentum:

General spin Hamiltonian:

relaxationcontribution

dephasingcontribution

Golovach, Khaetskii, DL, PRL 04


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$$: Swiss NSF, Nano Center Basel, EU (RTN), DARPA & ONR, ICORP-JST

Spin Qubits in Quantum Dots

Part II

Daniel Loss

Department of PhysicsUniversity of Basel

Switzerland


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All-electrical control and read-out achieved

Spin qubits in GaAs dots present status

1-qubit gate electron spin resonance

gate duration ~ 25 ns; observed 8 periods

2-qubit gate exchange interactiongate duration ~ 0.2 ns; observed 3 periods

Phasecoherence

T2* ~ 20 nsT

2

> 1 s

See also Hanson et al., Rev. Mod. Phys. 2007

Initialization 1-electron, low T, high B0

Read-out via spin-charge conversion

Energy

relaxation

T1~ 1 secduration ~ 100 s; 82-97% fidelity

duration ~ 5 T1; 99% fidelity ?


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Outline

A. Spin qubits in quantum dots

1. Basics of quantum computing and quantum dots

2. Quantum gates: interaction based and measurement based entanglement

B. Spin decoherence in GaAs quantum dots

Spin orbit interaction and spin decay Alternative spin qubits: holes, graphene, nanotubes,... Nuclear spins and hyperfine induced decoherence

classical spin bath vs quantum and non-Markovian behavior

C. Nuclear spin order in 2DEGs (ultra-low temp.)

Kondo lattice in marginal Fermi liquid


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Spin orbit interaction in GaAs quantum dots

interaction between spin and charge fluctuations SOI is weaker in quantum dots than in bulk since dot = 0

(Khaetskii & Nazarov, `00; Halperin et al., `01; Aleiner & Falko, `01)

e.g. spin-phonon: Khaetskii & Nazarov, PRB 64 (2001)Golovach, Khaetskii & Loss, PRL 93 (2004)

Fal'ko, Altshuler, Tsyplyatev, PRL (2005)Bulaev & Loss, PRL `05 (hole spin)

( ) ( ) )( 3pOppppH yyxxxyyxSO +++=

Dresselhaus SOIRashba SOI


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Basics on Spin-Orbit Interaction

Relativistic (Einstein) correctionfrom Dirac equation:

=m

pVs

mcHso

rr

22

1

MeV12 2 mc

p

( )p

free electrons

eV1gE

k

( )k

semiconductor (Zinc-blende)

conduction band

heavy holes

light holes

bandstructureeffects

R hb i bit i t ti i 2D


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Rashba spin-orbit interaction in 2D

spintronics

Rashba & Dresselhaus spin-orbit in 2DEG


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E. Rashba, Nature Physics 2, 149 ('06)

Rashba & Dresselhaus spin orbit in 2DEG


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Rashba & Dresselhaus Spin-Orbit in 2DEG

( )xyy

xR ppH =

( )yyx

xD ppH =

zE~Structure-inversion asymmetry tunable bygates: (Rashba, 1960)

Bulk-inversion asymmetry (Dresselhaus, 1955)


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Rashba & Dresselhaus Spin-Orbit in 2DEG

( )xyy

xR ppH =

( )yyx

xD ppH =

zE~Structure-inversion asymmetry tunable bygates: (Rashba, 1960)

Bulk-inversion asymmetry (Dresselhaus, 1955)

=

is conserved spin (new basis) which makesspin of electrons independent of their momentum ( spin FET ?

tune Rashba term s.t.

yxDR ppHH m=+ ( )yx

( )yx

Spin-Orbit Interaction in GaAs Quantum Dots (2DEG):


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( )xyyx

yyxxSO

pp

ppH

++=

p ( )

Dresselhaus S.-O.

Rashba S.-O.

( )tUHHHH phelSOZdot +++=

Model Hamiltonian:

( )rUm

pHdot += *

2

2

B = BZ gH 21

( ) ...= tU

phel any potential fluctuation, e.g., phonons

( )2

r

*

2

~m

h

piezoelectric & deformationacoustic

Electron-phonon interaction (2D):


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Electron phonon interaction (2D):

( ) ( )( ) += +

j

jjjj

qjc

i

zphel bbiqeeqFU

, /2

q

qqqq

rq

h

zq,qq=

( )( ) ( )( )( )qqqjj

j eqeqqq

+=

2

2

piezo-electric interaction:

( )

( ) ( )

( )( )qqqjj

j eqeqq

+= 2

1

deform. potential interaction:

for GaAs:1,0 jj =q and

=

=otherwise,0

)(,14 cyclicxyzh

Parameter regime:


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Parameter regime:

*, mSOSO h=


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Parameter regime:

In this regime, we find that

12 2TT =

--- the spin-orbit interaction in GaAs quantum dots causes

a spin decay with the largest possible decoherence time,T2=2T1.

Note that, generally, T22T1, and, usually, one expectsT2


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Parameter regime:

(typically: 100 nm, and SO1-10 m

2*2 mTkB h


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( )tUHHHH phelSOZdot +++=( ) ( )[ ]tUStUHHHeeH phelphelZd

SS

+++= ,

~

[ ] SOZd HSHH =+ ,with S defined bywith px=im*[Hd,x], get [ ] == ddSO LiHiH ,

( ) ( ) ......

1

1 102

++=

+=

+= SSLi

L

L

LLi

LLS d

d

Z

d

d

Zd

no orbital B effect to O(Hso):

( ) ,10 = iPS nEnH nd =, =n

nn nnAAP

( )

( ) ( )[ ] 0,,0

== nnphelnnphel tUitUS

1st order in HSO

leading order is due to Zeeman term (no orbital):


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( )( ) ,2

1

eff BB += tgH

B

( ) ( ),2 tt BB =

( ) ( ) ,/, 1 SOpheld tULt =

( )

,

1

11

=

= B

d

BZ

d L

PgiLL

PS

constanttindependenspin~

eff += HH

giving the effective Hamiltonian:

where

( ),0,',' += xy ( ) ,1*

h = m( )

( )

=

+=

2'

2'

yxy

yxxwith

Linear SO & Zeeman transverse fluctuations only:


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( ) ( ) ''' aa +== teHetH tiHSOtiHSOZZ

,, ba == ZSO HH

( ) ( ) ( )[ ][ ]nanaaa +=+= tttHSO '''''

bbn=( ) ( ) ,''' aaannnana +=

first term commutes with HZ

( ) ( ),'''' tt aa =since rotation is around n.with ( ) ,0'' =na t

i.e. Zeeman interaction induces only

fluctuations transverse to b

( ))( tpaa=

go to rotating frame:

)('' tb

Now:

)('' tb

'''' aa RUU =+


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1st order in Hso (time dependent perturbation theory):

( ) ( )[ ][ ]nan + tdttxdt

ddtctHdt

TT

SO

T

'')(000

SO linear : a ~ p ~ dx/dtnot a total derivative

for (He-ph)n, n>0

const. no orbital B field effect *)

Thus: if SO interaction linear in pwe get only transverse

fluctuations no pure dephasing T2=2T1

*) note: no orbital B field effect--only Zeeman, see alsoKhaetskii & Nazarov 00, Halperin et al., 01, Aleiner & Falko, 01

Derive Bloch equation for spin dynamics:


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SSBS += Bg&

( ) ( ) ( )

=

0

2

22

02

dtetBBg

wJ iwt

jiB

ijij

h

spectral function

( ) ( ) ( ) ,Re wJwJwJ ijijij = ( ) ( ) ( )wJwJwI ijijij =

Im

relaxation:

dephasing:

decay: ,dr +=

( ) ( ) ( ) ( ) ( ) ( ), ++

+= qjpipqpqkkpqijpjpiippqqppqijr

ij IlIlJllJll

( ) ( )00 ++ = pjpipqqpijd

ij JllJll 0

c=/ s=100 ps


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quantum well piezo deformation

( ) ( )( )

( )SOj

j

j

j

sz

j jc

zXX

sze

szFe

dsm

NzzJT

j 222

2

222

2

2sin

2

0

3

522

0

*

32

1

/cos

sin2

12Re1

22

+

+=

+

h

22

22

2

2

'

2

2

'

2

2

'

2

2

' 411112

+++

+

+

=

zyxyx lllll effective SO length

Bgz B=Golovach, Khaetski, Loss, PRL 93 (2004)

Relaxation rate:Bose function super-Ohmic: ~z

3


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smsssms /1037.3,/107.4 3323

1 =

quantum well piezo deformation

( ) ( )( )

( )SOj

j

j

j

sz

j jc

zXX

sze

szFe

dsm

NzzJT

j 222

2

222

2

2sin

2

0

3

522

0

*

32

1

/cos

sin2

12Re1

22

+

+=

+

h

13,/16.0,sin)1cos3(23

,2sin2,cossin23,7,

2

14

21

14

2

1

14

221

14

2

001,

2

,3

,2,1

=

===

mChh

hheVjj

22

22

2

2

'

2

2

'

2

2

'

2

2

' 411112

+++

+

+

=

zyxyx lllll effective SO length

speed of sound

Bgz B=

Spin relaxation rate 1/T1 for GaAs Quantum Dot


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( ) 22 2sin2

0

122

1

sin)(1

jB sBgk

B edBgT

++

22 BB 2

)( ph pHSO

power-5 law for B< 3T (GaAs)

Golovach et al., PRL 93 (2004)

Numerical value of T1 for GaAs parameters (13!):


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)()0020.00077.0(04.043.0 TBg =Hanson ea PRL 91 (196802) 200329.0=gor with linear fit:

Theory: T8Bfor,s7501 =T

( )

===

e

cSO

m

mssshmd

067.0,mkg103.5,scm1035.3,scm1073.4

,mC16.0,eV7.6,1.13,0,m9,nm5,nm32,meV1.1,,,,,,,,,,,

3355

2

*

321140

*

0

hh

Zumbuhl ea PRL 89 (276803) 2003

TsT [email protected] =Experiment:Elzerman et al.,

Nature 430, 431 (2004)

22/SO

s11005501 =T due to uncertainties in gfactorms7.21=T for SO=17m [Huibers ea PRL 83, 5090 (1999)]

Current record: T1 > 1 s (B 1 T) in GaAsS A h K M L I R d D Z b hl


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S. Amasha, K. MacLean, I. Radu, D. Zumbuhl,

M. Kastner, M. Hanson, A. Gossard, Phys. Rev. Lett. 100, 46803 (2008).

data in good agreement with theory

Golovach, Khaetskii, DL, PRL 93 (`04)

Rashba & Dresselhaus SOI Effectswell understood (?)



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( ) 22 2sin2

0

5

1

sin)(1

jB sBgk

B edBgT


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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

1

2

3

T

1

[ms]

S-T splitting [meV]

Theory: Golovach, Khaetskii, Loss, PRB 2007

Experiment on double-dots, Meunier et al., PRL 2007

phonon wavelength

matches dot size

p 1


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Brute force nuclear polarization with large B-field:

Note: spin phonon rate is non-monotonic function of B-field

Chesi & DL, cm0804.3332

1/T1,2 depends strongly on B-field direction (magic angles):


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( )( )0,2/

,,1

11 ===

T

f

T

( ) ( )( )[ ]

2sinsin2cos11

,, 22222

+++=f

43,2, === 1Texact!

Special case:

Rashba and Dresselhaus interfere! *)

*) Schliemann, Egues, DL, PRL `03

Golovach, Khaetskii, LossPRL 93, 016601 (2004)

ellipsoid

1/T1,2 depends strongly on B-field direction(magic angles):


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( )( )0,2/

,,1

11 ===

T

f

T

( ) ( )( )[ ]

2sinsin2cos11

,, 22222

+++=f

43,2, === 1Texact!

Special case:

Rashba and Dresselhaus interfere! *)

*) Schliemann, Egues, DL, PRL `03

Golovach, Khaetskii, LossPRL 93, 016601 (2004)

ellipsoid

Relaxation of spin in GaAs quantum dots dominated by


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spin-orbit & phonons with ultra-long relaxation times T1:

T1 ~ 1s for B ~ 1TAmasha et al.,Phys. Rev. Lett. (2008)



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e a at o o sp Ga s qua tu dots do ated by

spin-orbit & phonons with ultra-long relaxation times T1:

But measured spin decoherence times are muchshorter: T2~1- 10 s Petta et al. '05; Koppens et al. '06/'07

T1 ~ 1s for B ~ 1T

From Rashba- SOI we expect T2 = 2T1 Golovach et al., PRL '04

Thus, spin decoherence in GaAs must be dominated byother effects hyperfine interaction with nuclear spins

Burkard, DL, DiVincenzo, PRB 99

Amasha et al.,Phys. Rev. Lett. (2008)

Hyperfine interaction:


68/157

Major source of dynamics/decoherenceQuantum dots Si:P donors

Molecular magnetsNV centers in diamond

Ono and Tarucha, PRL (2004),Petta et al., Science (2005),

Koppens et al., Nature (2006)E Abe et al., PRB (2004)

Childress et al., Science (2006),Hanson et al., PRL (2006)

Ardavan et al., PRL (2007)Barbara et al., Nature (2008)

...All are potential candidates for quantum information processing applications


69/157

1. Avoid nuclear spin problem: use holes or othermaterials such as C, Si, Ge,...

2. Use GaAs (still best material) and dealwith nuclear spins

Strategies:


70/157

1. Avoid nuclear spin problem: use holes or othermaterials such as C, Si, Ge,


Strategies:

Heavy Hole Spins in Quantum Dots

fl d HH LH i i d l li d HH


71/157

Heiss et al., Phys. Rev. B 76,2413062 (2007);

[and: Gerardot et al., Nature451, 44108 (2008)]

flat dot HH-LH mixing suppressed long-lived HH

Bulaev & Loss, PRL 95, 076805 (2005)

Abstreiter/Finley group 07: T1 ~ 200 s

self-assembled InGaAs dots

Hyperfine effect on holes??

Hyperfine Interactions with Nuclear-Spins


72/157

Hyperfine Interactions with Nuclear Spins

Fermi contact hyperfine interaction

Anisotropic hyperfine interaction

Coupling of orbital angular momentum

kkjS

k

ISrh k

rrr

= )(38

4

01

)/1())((3

4 30

2

kk

kkkkjS

k

rdrISInSnh

k +=

rrrrrr

)/1(43

03

kk

kkjS

k

rdr

ILh

k +

=

rr

Band structure of a GaAs QW


73/157

Band structure of a GaAs QW

In the quasi-2D limit, a gapdevelops between the HH andLH sub-bands.

The HH-LH degeneracy islifted.

For GaAs:

eV

eVeVE

LH

SO

g

1.0

3.05.1

=

==

(QW height of5nm)

s-type

p-type

Hyperfine Interactions with Nuclear-Spins


74/157

yp p

Fermi contact hyperfine interaction

Anisotropic hyperfine interaction

Coupling of orbital angular momentum

kkjS

k

ISrh k

rrr

= )(38

4

01

)/1())((3

4 30

2

kk

kkkkjS

k

rdrISInSnh

k +=

rrrrrr

)/1(43

03

kk

kkjS

k

rdr

ILh

k +

=

rr

Electronsh1 isotropic

Holesh2,3 Ising-like

h2,3 ~ 0.2 h1

Fischer, Coish, Bulaev, Loss, arXiv:0807.0368

Carbon-based Nanostructures


75/157

Folklore: 12C is light atom and thus weak spin-orbit interactionis it true?

)( 3 yyxx ppvH +=

2D Dirac-Hamiltonian:

Spin Qubits in Graphene


76/157

Trauzettel, Bulaev, Loss & Burkard, Nature Physics 3, 192 (2007)

Advantages: weak SOI and

(almost) no nuclear spins;& long-range interaction

Challenge: armchair boundariesto lift orbital degeneracy

Carbon Nanotube Quantum Dots


77/157

Nygard, Cobden, Lindelof, Nature 408, 342-6 (2000).Jarillo-Herrero, et al., Nature 429, 389 (2004).Mason, et al., Science 303, 655 (2004).Graber, et al., Phys. Rev. B 74, 075427 (2006).

Carbon Nanotube:


78/157

Carbon Nanotube:

Dirac-like bandstructure withsmall gap: semiconducting NT:

Gate-controlled confinment:

Spin orbit interaction: zparaiperp

SOI ScheSiH 132 2.).( ++= +

Ando, J. Phys.Soc. Jpn., 2005;

Huertas-Hernando et al., PRB 2006

Spectrum of Nanotube Quantum-Dot in B-fieldD. Bulaev, B. Trauzettel, and D. Loss, PRB 77, 235301 (2008)


79/157

Spin-Orbit Interaction leads to zero-field splitting



80/157

Spin-Orbit Interaction leads to zero-field splitting

Spectrum experimentally

confirmed, see Kuemmeth, Ilani,Ralph, McEuen, Nature 452 (2008)



81/157

Spectrum experimentally

confirmed, see Kuemmeth, Ilani,Ralph, McEuen, Nature 452 (2008)

Spin-Orbit Interaction leads to zero-field splitting all-electrical control of spin possible!

Spin Relaxation and Spin Decoherence Rates in Nanotubesdue to Electron-Phonon and Spin Orbit Interactions


82/157

Extreme variations withB-field: spin relaxation rate

can be varied by a factor 108

!

1/f phonon noise

due to quadratic bending modes ~ q2

in 1D

Bulaev et al., PRB 77, 235301 (2008)


83/157

1. Avoid nuclear spin problem: use holes or othermaterials such as C, Si, Ge,


Strategies:

Spin-bath dynamics


84/157

Fully-polarized bath exact dynamics

Thermal (completely random) bath

Spin-echo decay

Semiclassical dynamics

Khaetskii, Loss, Glazman, PRL (2002), ...: power-law or inverse-log decay

Khaetskii, Loss, and Glazman, PRL (2002), Merkulov, Efros, andRosen, PRB (2002), Coish and Loss (2004), ...: Gaussian decay

Erlingsson and Nazarov, PRB (2004), Yuzbashyan et al., J. Phys. A.(2005), Al-Hassanieh et al., PRL (2006), Chen, Bergman, and Balents,PRB (2007), ...: power-law or inverse-log decay

de Sousa and Das Sarma, PRB (2003), Witzel and Das Sarma, PRL(2007), Yao, Liu, and Sham, PRL (2007), ...: super-exponential decay

BUT: Non-exponential decay inconsistent with Markovianquantum error-correction models!!

zvv

Hyperfine Interaction in Single Quantum Dot


85/157

x

y

zB( )rr

z

( ) nKsHdd 1001014

2/12

nuclear spindipole-dipole interaction

ddB

i

ii HSBgISAH ++=

electron Zeeman energy

2)( ii rAA

v

hyperfine interactionis non-uniform:

516 10,10 = NsN

A

Khaetskii, DL, Glazman, 02; Coish & DL, 04-07; Eto 04; Sham 06; Altshuler 06;Balents 07 ...

Separation of the Hyperfine Hamiltonian


86/157

Hamiltonian: VHhSBSgH zB +=+= 0vv

= iiiIAh

vv

Note: nuclear field

... ... ... ...

zzB ShBgH )(0 +=

( )++ += ShShV

2

1

longitudinal component

flip-flop terms

Separation:

yx ihhh =V V

is a quantum operator

Nuclear spins provide hyperfine field hwith

( ) fl i b l i


87/157

(quantum) fluctuations seen by electron spin:

Sv

hv


( t ) fl t ti b l t i


88/157


Sv

hv


( t ) fl t ti b l t i


89/157


With mean =0 and (quantum) varianceh:

1

2

1

2

)10(5/

= ===

== nsmTNAIAhh nucl

N

k

kknucl

r

what state?

Svh

v

Initial conditions for nuclear spinsCoish &DL, PRB 70, 195340 (2004)


90/157

[ ] === knnz

z

kkzI nhnIAnhnn ,)0(

)3(

Pure state of hz-eigenstate:



91/157

( )

( )

=

=

+==

N

NNNI

k

i

k

N

kIIII

NNffNN

fef k

1)0(

1,)0(

)2(

1

)1(

[ ] === knnz

z

kkzI nhnIAnhnn ,)0(

)3(

Superposition (1) or mixture (2) of hz-eigenstates:

Pure state of hz-eigenstate:



92/157

[ ] === knnz

z

kkzI nhnIAnhnn ,)0(

)3(

( )

( )

=

=

+==

N

NNNI

k

i

k

N

kIIII

NNffNN

fef k

1)0(

1,)0(

)2(

1

)1(

Superposition (1) or mixture (2) of hz-eigenstates:

Pure state of hz-eigenstate: Pure state of hz-eigenstate:

nuclear spin bath behaves classically !

Gaussian decay and narrowing

Zeroth order in flip-flop terms: )]0([Tr )( thBgi zBeSS +


93/157

Zeroth order in flip-flop terms:

Gaussian decay

)]0([Tr )(0 I

thBgiIteSS +++ =

,22 2

0

.)(cttti

measno

t eeSS

++

zh

h

B

nsp

N

Atc 5

1

22

= h

GaAs, N=105

C h t i l t t i l t

(tunneling)

E. Laird et al., PRL '06: free induction decay


94/157

Theory (full lines): Coish & DL,PRB 72, 125337 (2005)

Coherent singlet- tripletoscillations due to hyperfinemixing with Gaussian distribution reasonable agreement

between theory and experiment

Koppens et al., PRL 2007: driven Rabi oscillations


95/157

Gaussian Hyperfine field :

(Klauser, Coish & DL)

Gaussian decay and narrowing

Zeroth order in flip-flop terms: )]0([Tr )( IthBgiIzBeSS +++ =


96/157

Zeroth order in flip flop terms:

Prepare nuclear spin system with a measurement of the Overhauser field

B. Coish and D. Loss (PRB 2004), G. Giedke et al. (PRB 2006)D. Klauser et al. (PRB, 2006), D. Stepanenko et al. (PRL 2006)

Gaussian decay

)]0([Tr0 IIt

eSS ++ =

,22 2

0

.)(cttti

measno

t eeSS

++

zh

h

B

Precession

[ ] nnzBtimeas

t hBgeSS += ++

,0

.)(

B

h

nsp

N

Atc 5

1

22

= h

GaAs, N=105

Narrowing of nuclear spins in double dots with ESR

Klauser, Coish & DL, PRB 73, 205302 (2006)

ESR: oscillating exchange J(t) J +j cos(t) leads to Rabi oscillations:


97/157

z

nB hBg +

ESR: oscillating exchange J(t)=J0+j cos(t) leads to Rabi oscillations:

+=

=

ESR at frequency = measures eigenvalue nuclear spins projected into corresponding eigenstate |n>

z

nB hBg + z

nh

If quantum measurement is ideal, then Gaussian superpositioncollapses to a single Lorentzian (ESR linewidth):

j/2

Quantum control of


98/157

0

0.2

0.4

0.6

0.8

1

1.2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100

b)a)

c)

M = 51, = 1

M= 100, = 22

M= 50, = 0

(x x0)/0

(x x0)/0

(x x0)/0

M

d)

(x) (x)

(x)P

Quantum control ofnuclear spin bath

through measurementof single electron spin

Klauser, Coish & DL, PRB 73, 205302 (2006)


99/157

[ ]

===

knnz

z

kkzI

nhnIAnhnn ,)0()3(

Consider now nuclear spins in pure state of hz-eigenstates:

hv

Narrowed initial nuclear spin state h=0 at t=0


100/157

hSv .constIS

k

k=+rr

Khaetskii, DL, Glazman, PRL 02 & PRB 03

Coish &DL, PRB 70, 195340 (2004)

t>0: quantum dynamics ... ... ... ...V V

changes hyperfine field in time by 1/N spin precesses influctuating hyperfine field spin dephases (power law decay)

flip-flops

back action of Son hBz

hv

IS

Narrowed initial nuclear spin state h=0 at t=0


101/157

hSv

Dynamics (flip-flops): ... ... ... ...V V

.constISk

k=+rr

2/3

/

2

2

)/()(4

)0()(

NAt

e

pIAbN

AStS

NitA

zz

+

Time scale is N/A = 1s (GaAs) and decay is bounded

power law decay

back action of Son h

E.g.

Bz

Khaetskii, DL, Glazman, PRL 02 & PRB 03, Coish &DL, PRB 04

Quantum vs. Classical Dynamics


102/157

N

A

IKKSBSH

N

kk

z

==+= = ;; 1vvv

Special case: Uniform hyperfine coupling =kA

)()()(

)()()(

tstktk

tstkBts

vv&vv

vv&v

=

+=

Classical: Time-dependentmean-field (TDMF)

Exact solutioncf. Yuzbashyan and Altshuler,

J. Phys. A (2005)

Quantum:

Exact solutioncf. Coish, Yuzbashyan, Altshuler, Loss

J. Appl. Phys. 101, 081715 (2007);

[ ][ ]KHiK

SHiSv&v

v

&

v

,

,

==

t

t

K

Sv

v

Initial Conditions

Classical Quantum


103/157

( )

( )sssss

kk

s

k

cos,sinsin,cossin)0(

cos,0,sin)0(

=

=

v

v

( )==

+=

=

m

k

K

mK

iK

k

siss

ks

mKdKKe

e

ky

s

,,2

sin

2

cos

)0(

)(

Classical Quantum

)0(sv

)0(kv

B

Sv

Kv

B

classical vectors spin coherent states('minimum uncertainty')

( )0=B

Classical vs. Quantum Dynamics


104/157

( )

( ) N

AT

tsS

tsStd

TtC

xtx

xtxTt

t

==+

=

+

;1.0

;21

)(22

( )

( )10=B

TDMF:

quantum:

)(tsx

txS

Time-dependent mean-field (TDMF) = exact quantum solution1)( =tC

)(tsS xtx for times t> AN

c

2=1)(


105/157

Time

21

~

b

A

N

A

Nc ~

A

N

A

b~

???~2T 2Tt>>

tx

Fully polarized nuclei: exactly solvable case

BSIS AH

Khaetskii, DL, Glazman, PRL `02 & PRB `03


106/157

The initial nuclear spin configuration is fully polarized. With the initial wavefunction 0 we construct the exact wave function of the system for t > 0 :

Normalization condition is: , and we assume that

From the Schrdinger equation we obtain:

paramagnon

entangled

BSIS += Bii

i gAH

(1)


107/157

Laplace transform of (1) gives:

++

+

+=

++=+==

k

k

k

iu

iuuD

iut

uDi

uDtiu

2/A4/)2A(

A

4

1

4

)2A()(

,2/A4/)2A(

)0(A)(2)(

)0()(

kz

2

kz

kz

k

])2

)(A1(lnN2i-A[2

2/A

A 20

k

2

k =+

N

zidz

ik

Introducing 4/)2A( z ++= iiu , using (t=0)=1, k (t=0)=0,and replacing the sum over k by an integral

self-energy

(1)


108/157

where A = kA k; l=1,, N set of N+1 coupled differential eqs. (N>>N)

Spin correlator: 2/))(1()()(2

000 tStStC zz ==

Laplace transform of (1) gives:

here

note: sums k replaced by integrals over rk3 (valid for t < N2/A),

with x,y (Gaussians) integrated out non-analyticity

integration contour and singularities


109/157

The singularities are: two branch points (=0, 0= i A02(0)/ 2N), and firstorder poles which lie on the imaginary axis (one pole for z > 0, two poles for

z < 0). For the contribution from the branch cut (decaying part) we obtain:

)0(20 =here and .)0()(),(2

00

2

000 == zzz

g g

1) Large Zeeman field |z| >> A :

Asymptotic behavior ( = t/(N/A) >>1) determined by nuclei at dot center:


110/157

i.e. power law decay and bounded by 1/N and (A/z)2

The decay law depends on the magnetic field strength . However, the

characteristic time scale for the onset of the non-exponential decay is the same

for all cases and given by (A/N )-1 (microseconds in GaAs dot).

~ 1/ N

2) No external magnetic field (z= 0):Asymptotic behavior ( >> 1) determined by weakest coupled nuclear spins:

2/3ln/1~ non-universal

z

Bi

ii SBgISAH +=

vv

General case: below full polarization


111/157

x

y

zB

( )rr

z

Coish & Loss, 04-07; Eto 04; Sham 06; Altshuler 06; Balents 07,...

Generalized Master Equation

Generalized Master Equation (exact): [ ]OHLO ,=

Bill Coish &DL, PRB 70, 195340 (2004)

Liouville op.


112/157

tdtttitiLt

t

SSSS =

0

0 )()()()( &

)0()( IViQLt

IS LLeiTrt =

[ ] )0()()( 1)2(0)2( SSS siiLss

++=

Ge e a ed aste quat o (e act)

L++=+= )()()0(1

)()4()2(

ssLiQLsLiTrsSSIVIS

)0(11

)0(1

)0(1

00

IVVIIVIIVI LiQLs

QLiQLs

LTrLiQLs

LiTrLiQLs

LiTr ++

+

=+

Lowest-order Born approximation:

+

=i

i

S

st

iS dsset

)()( 21

Laplace transform, expand self-energy in the transverse coupling :VL

)()( tTrt IS =

[ ]

[ ]OVOL

OHOL

V ,

,00

=

=

( )OTrQO II )0(1 =

)0()0()0( IS =

nnI =)0(

[ ] nhnhnnzz

=

Initial conditions and perturbative regime

Initial State: )0()0()0( IS =1

El i[ ]=

nnzznhnh ,


113/157

j

z

jI mnInnn == ,)0(

zzyyxxSS SSSI 0002

1)0( +++=

Nuclear spins:

Electron spin:

=i

z

iiz

nnzz

IAh

Initial conditions and perturbative regime

Initial State: )0()0()0( IS =SSSI

1)0(El t i

[ ]=nnzznhnh ,


114/157

)(

)()( 0

sis

sNSsS

zz

zz

z ++=

)()( 0

siis

SsS

n ++

++ +=

longitudinal and transverse electron spin dynamics decouple to all orders:

j

z

jI mnInnn == ,)0(

zzyyxxSS SSSI 0002

1)0( +++=

Nuclear spins:

Electron spin:

=i

z

iiz IAh

[ ]nnzn

hb+=

Perturbative regime for all electron spin dynamics given by: IAb >>

( )1

12

0

2/1

0

2


115/157

0iLs +

[ ]

[ ]

[ ])()()(

)()()(

)()()(

)2(

)2(

)2(

sIcsIciNs

isIisIiNcs

isIisIiNcs

nn

nn

++++

+

++

+=

++=

++=

1,ln

2

4

1)(

1

0

2

== md

ixs

xxdx

m

d

Ais

A

NsI

k k

k

mm

=

m

l

rr

02

1exp)0()(

In d Dimensions:

),1:2/1(

)()()(

,)1()1(

+

===

=

+=

fcfcI

mFmPmF

mmIIc

m

I

d=m=2: [ ] isisssI = )log()log()( mbranch cuts

Dependence on the envelope wave function

1

Donor Impurity:

( )02/exp)( lrr

1exp)0()(

=m

rr

Envelope wave function:


116/157

0 10 20 30 40 50

t [2 --h/A0]

-0.5

0

0.5

Re[I

+(t)/I

0]

2D Quantum Dot:

[ ]sANt 1/2 h

( ) 2//exp)( 20lrr

)0(

)(

+

+

R

tR

.2

exp)0()(0

=l

r

,2,

1

)1(/

> m

d

ettR NiAt

m

d

X

,11,ln

)1(2

=>>

m

d

t

ttRX

in d dimensions.

Decaying fraction of the spin:

zXSStR XtXX ,,)( 0 +==

2

2

)(4 pIAbN

A

+=

Dot:

Si:P

Spin dynamics is `non-Markovian (for t~N/A)

Born-Markov Approximation not valid:


117/157

( )

t

ti

t

t

t

tti

ttSeSStttdiSiS nn +

+++++

++ ==

~

0

)2()( ,)(e~&

( )

( ) !0)~(Im

)~(Re~

)2(

)2(

=+==

+==

++

++

n

n

is

is

)(0

tRSeS t

t ++

+ +=

From the exact GME:

i.e. vanishing decay rate!

Markov approximation(in the rotating frame)

non-Markovianremainder term

0,1

)0( *

2

=+ zBBgNp

R

Stationary Limit & Decaying Fraction

p

Longitudinal spin: Stationary limit within the Born approximation (I=1/2):


118/157

+

+===

12)(lim

1lim

0

0

0

pS

ssSdtST

Sz

zs

T

tzTz

Np21

zS

Decaying fraction for b=0 is of order independent ofdimensionality or specific form of the electron envelope wavefunction.

This result is confirmed by exact diagonalization ofsmall systems, e.g., Ntot=15:

0 5 10 15 20t (2N/A)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

t

)2

1,0(

122

==== IbNp

N

n

Summary of electron spin dynamics

zXSStR XtXX ,,)( 0 +== 2*2

)(4)(

pIABgNAtR

zB

X +=


119/157

=

m

l

rr

02

1exp)0()( in d dimensions, and for a wave function:

+= 1

* pIABg

A

zB

Exact when p=1,Born approximation valid in

the weakly perturbative regime.


120/157

Time

21

~

b

A

N

A

Nc ~

A

N

A

b~ ???~

2

T2

Tt>>

tx

Free-induction decay: Big picturePower law:

Khaetskii, Loss, Glazman, PRL (2002)Coish and Loss, PRB (2004)

tx


121/157

Time

21

~

b

A

N

Initial quadratic (Gaussian?) decay:Yao, Liu, Sham, PRB 06, PRL 07, NJP 07

2)/(2)/(1 tt etx

A

Nc ~

A

N

A

b~ ???~

2

T2

Tt>>

t



tx


122/157

Time

???

21

~

b

A

N


2)/(2)/(1 tt etx

A

Nc ~

A

N

A

b~ ???~2T 2Tt>>

t



tx


123/157

Time

???

21

~

b

A

N


2)/(2)/(1 tt etx

A

Nc ~

A

N

A

b~ ???~2T 2Tt>>

Long-time power law:Deng and Hu, PRB (2006)

2/1 tx t

t

Consider now times t >> N/ACoish, Fischer, and Loss, Phys. Rev. B 77, 125329 (2008)

Coherence of an electron spin can exhibit exponential


124/157

p pdecay for large magnetic field and narrowed initial

bath-state Hyperfine Hamiltonian

2nd order effective Hamiltonian (Schrieffer-Wolff trafo)

Reproduces 4th-order rates when expanded to 2nd order in flip-flopsonly

)(2

1)( ++ ++++= ShShIbShbH

k

z

kk

zz

hf

+

+++=

+

k

z

kk

z

lk

lkz

lkz IbSIIhb

AAhbH

2

1

Details of the Transformation

Write the hyperfine Hamiltonian as

( )++ +++ ShShVSHVHH z 1


125/157

Schrieffer-Wolff transformation

This can be rewritten as

( )+=+=+= ShShVSHVHH zhf2

,, 000

VL

SVSHeHeH S

hf

S

0

0

1],,[

2

1=+=

( ) DSXSSHSSHH z ++=+= ++

diagonal off-diagonal

(with respect to product eigenbasis of )z

kI

Hyperfine-Mediated Nuclear-Field Fluctuations Heff

)()( 3VODSXH zeff +++=


126/157

Effective (fluctuating) field leads topure dephasing of electron spin

Fixed frequency(narrowed initial state)

(small for large b)

= 10],[ TSH z

Beff ~ +X

2/Tt

tx eS

zhb +

z

k

k

kID

bIIAA

X lklk

lk

1

2

1

+

Generalized Master Equation (GME)

Initial conditions:

nnnn === )0()0()0()0(


127/157

Nakajima-Zwanzig GME:

Expand in powers of

nnnn nIIs === ,)0(,)0()0()0(

'0

)'('t

t

tn

t

SttdtiSiSdt

d+++

=

bXSV z 1~=

...)()()()4()2(

++= ttt 1


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Markovian regime:

+ 0 )(Re, tSex tt

)()(

~ )(tet

ti n

= +

2Tc


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Markovian regime:

+ 0 )(,tt

)()(

~ )(

tet

ti n

= +

2Tc


130/157

Markovian regime:

+ 0 )(,tt

)()(

~ )(

tet

ti n

= +

2Tc


131/157

Homonuclear system:

Continuum limit:

=

02

)0()(Re1

XtXdteT

ti titiXeetX

=)(

tAAi

l

lk

klkeAA

bXtX

)(22

2

1)0()(

tAAi

lklkeAAdldk

b

)(22

002

1 ANc /

)0()( XtX

t

Distribution of coupling constants

k spins

N spinsd=1

(e.g. nanotube, nanowire)


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d=2

d=3

volume occupied by one spin

(e.g. lateral quantum dot)

(e.g. Self-assembled dot, defect)

Decoherence Rate:

Homonuclear System


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Coupling toone nucleus

Geometricalfactor

N

A

b

A

q

df

II

T

22

2 3

)1(1

+=

4~I 1


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Markovian

Non-Ma

rkovian

1D quantum dot (e.g. nanotube, nanowire)

donor impurity

2D quantum dot(e.g. lateral gated dot)

2T

=

q

Ba

rr exp)0()(

Heteronuclear system

(neglecting inter-species flip-flops*) =

i

iiT

2

2

1


135/157

Isotopic abundance

IIn =9/2 IGa,As =3/2

InxGa1-xAs, x=0.05

*) if difference in nuclear

Zeeman energy > A/N

explains strong isotopedependence seen in transport

measurements (Ono+Tarucha,04; Yacoby 07)

Free-induction decay due to HyperfinePower law:


2Long-time power law:

tx


136/157

Time

???

21

~

b

A

N


2)/(2)/(1 tt etx

A

Nc ~

A

N

A

b~ ???~2T 2Tt>>

g pDeng and Hu, PRB (2006)

2/1 tx t

Power law:Khaetskii, Loss, Glazman, PRL (2002)Coish and Loss, PRB (2004)

2

Long-time power law:tx

Free-induction decay due to Hyperfine


137/157

Time

21

~

b

A

N

Initial quadratic decay:Yao, Liu, Sham, PRB 06

2)/(1 tx t

A

Nc ~

A

N

A

b~ NbT

2

2

2 ~ 2Tt>>

g pDeng and Hu, PRB (2006)

2/1 tx tExponential decayCoish et al., PRB (2008)

2/Tt

t ex

Summary of Part A&B

spin qubits and entanglement: all-electrical control spins in GaAs dots: long relaxation times (secs!)


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spins in GaAs dots: long relaxation times (secs!)

due to SOI & phonons but short decoherence due to hyperfine interaction alternative systems: holes, graphene, nanotubes,...

power law decay for times up to N/A exponential decoherence with strong isotope

dependence ( for t > N/A, and large B)

1. Dynamical polarization

C. Polarization of nuclear spins


139/157

ESR & optical pumping:


140/157

A ~90eV in GaAs

A/EF

~ 10-2


141/157

Schrieffer-Wolff transformationintegrate out electron degrees of freedom

RKKY interactionbetween nuclear spins

(overrules direct dipolarinteraction)

electron spin susceptibility

A ~90eV in GaAs

A/EF

~ 10-2


142/157

Look at stability by considering spin wave excitations

electron spin susceptibilityfor noninteracting Fermi gas

2D

What is required for magnetic order?

Assume a ferromagnetic ground state(ensured by Weiss mean field theory)


143/157

Look at stability by considering spin wave excitationsspin wave excitation

spectrum

2D2D

What is required for magnetic order?

Assume a ferromagnetic ground state(ensured by Weiss mean field theory)


144/157

Look at stability by considering spin wave excitationsspin wave excitation

spectrum

continuum of zero energy excitations

nuclear order unstable no magnetization!

2D

But if...

Ferromagnetic order would be stable up to some T > 0if the spin wave dispersion was linear at |q| 0

Required: Non analytic behavior


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Required: Non-analytic behavior

Many-body corrections from electron-electroninteraction beyond standard Fermi-liquid theory?

Non-analyticities in Spin Susceptibility in 2DEG

Consider screened Coulomb U and 2nd order pert. theory in U:

)q(

,0 A typical diagram


146/157

Chubukov, Maslov, PRB 68 ('03)Aleiner, Efetov PRB 74 ('06)

,q


147/157

Possibility of spiral order

But it has the wrong sign!

Negative spin wave dispersion.


148/157

But is this generally true? Based on perturbation theory &screened Coulomb interaction

Shekhter & Finkelstein, PRB`06;

Braunecker, Simon, Loss, PRB08

Chesi, Zak, DL, in preparation

NO! Renormalization important in 2D!

Possibility of spiral order

Renormalization

Backscattering strongly renormalized by Cooper channelcontribution (P 0 & presence of Fermi sea).


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is possible

Leads to renormalization of scattering amplitude: () (,q)Saraga, Altshuler, Loss, Westervelt, PRB 05;Chesi, Zak, Loss, 08

Renormalize lowest order diagrams

Chesi, Zak, DL, in preparation


150/157

Chubukov, Maslov, PRB 68, 155113 (2003)

Renormalization

Include Cooper channel in S(q) (Kohn-Luttinger instability w.r.t. q):


151/157

renormalized scatteringamplitude

Saraga, Altshuler,Loss, Westervelt

PRB 05

dominating Cooper channel: = (: angle between incident electrons)

Since

if

for small |q|

Braunecker, Simon, Loss, PRB08

Ferromagnetic order is possible

Nuclear Ferromagnetstable


152/157

Also a possibility


153/157

Possibly spiral order

Critical Temperature

Nuclear Ferromagnetstable

shape obtained also


154/157

Finite magnetization at low temperatures -value of Critical Temperature?

shape obtained also

by Local Field Factorcalculation for

long-range Coulombinteractions

Critical Temperature

Nuclear magnetization at finite temperature


155/157

valid atwith the Curie Temperature

Finite magnetization at finite temperature in 2D!

Estimate for GaAs 2DEG: Tc ~ 25 K 1 mK, for rs ~ 0.8 - 8


156/157

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